# Full text of "Treatise On Analysis Vol-Ii"

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```2   REAL MEASURES       103

number a, £ 0 such that \^t(g)\ g at \\g\\  for all #eJf"(X;H,.) (13.1.2).
Hence, for each function /e jf (X; K), we have

so that

(13.1.10) If A and \JL are two measures on X, then so are A + ju and ak for
any scalar aeC. The set of all measures on X is therefore a vector subspace
of C*<*x\ which we denote by MC(X) or M(X).

By analogy with the example (13.1.4), if/x is a measure on a locally com-
pact space X, we write \fdfi or J f(x) d^(x) (or also </, /i> or </*,/» in place

of X/X f°r any/e Jf(X), and we call this number the integral off with
respect to /*.

2. REAL MEASURES

Let X be a locally compact space. Let ^TR(X) denote the set of all real-
valued continuous functions on X with compact support, and JfR(X; K) the
set of those whose support is contained in K. Clearly JfR(X) is a real vector
subspace of Jf C(X), and we can write

(direct sum). For every (complex) measure \JL, the restriction of ju to JTR(X)
is an R-linear mapping HQ of J>fR(X) into C; moreover u,0 determines ju
uniquely, for if/=/f + if2 with /ls/2 in Jf R(X),then /<(/) = A*oC/i) + ^oC/i)-
Conversely, if an R-linear mapping //0 : JfR(X) -> C is such that, for each com-
pact subset K of X, there exists % > 0 with the property that |/i0(/)l ^ % ll/ll
for all/e «yTR(X; K), then it is immediately obvious that the mapping

f   _i_ if  i_i. a   ( /* \ JL »

7l  "T" t/2 '—  rQ^Jl)   '   *J

is a (complex) measure on X. Hence we may identify each measure on X with
its restriction to Jf R(X).

Let fi be a (complex) measure on X. It follows immediately from (13.1.2)
that the mapping /i-»X/) ^ a'so a measure on X, called the conjugate of \JL
and denoted by /L We have Ji = ^, and if A, ^ are measures on X and a, b are
any two complex numbers, then ak = bfj. = ^1 4- 5/1. More generally, if # is
any function belonging to #cW and // is any measure on X, then we havesion" of v to X, if it exists,
```